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Properties

Label 2-930-465.464-c1-0-4
Degree $2$
Conductor $930$
Sign $-0.999 + 0.0349i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−1.70 + 0.288i)3-s + 4-s + (−2.10 + 0.760i)5-s + (−1.70 + 0.288i)6-s + 4.63i·7-s + 8-s + (2.83 − 0.985i)9-s + (−2.10 + 0.760i)10-s + 1.87·11-s + (−1.70 + 0.288i)12-s − 3.74·13-s + 4.63i·14-s + (3.37 − 1.90i)15-s + 16-s − 7.48i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.986 + 0.166i)3-s + 0.5·4-s + (−0.940 + 0.340i)5-s + (−0.697 + 0.117i)6-s + 1.75i·7-s + 0.353·8-s + (0.944 − 0.328i)9-s + (−0.664 + 0.240i)10-s + 0.564·11-s + (−0.493 + 0.0832i)12-s − 1.03·13-s + 1.23i·14-s + (0.870 − 0.491i)15-s + 0.250·16-s − 1.81i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.999 + 0.0349i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.999 + 0.0349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00887547 - 0.507453i\)
\(L(\frac12)\) \(\approx\) \(0.00887547 - 0.507453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.70 - 0.288i)T \)
5 \( 1 + (2.10 - 0.760i)T \)
31 \( 1 + (4.94 + 2.56i)T \)
good7 \( 1 - 4.63iT - 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 + 3.74T + 13T^{2} \)
17 \( 1 + 7.48iT - 17T^{2} \)
19 \( 1 + 5.69T + 19T^{2} \)
23 \( 1 - 7.09iT - 23T^{2} \)
29 \( 1 + 6.19T + 29T^{2} \)
37 \( 1 + 1.89T + 37T^{2} \)
41 \( 1 - 3.77iT - 41T^{2} \)
43 \( 1 + 5.39T + 43T^{2} \)
47 \( 1 - 6.33T + 47T^{2} \)
53 \( 1 - 2.24iT - 53T^{2} \)
59 \( 1 - 4.58iT - 59T^{2} \)
61 \( 1 - 7.47iT - 61T^{2} \)
67 \( 1 + 8.18iT - 67T^{2} \)
71 \( 1 + 14.8iT - 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 - 1.01iT - 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 1.87iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.88513086172265160487973101394, −9.627277186177788782312256430327, −9.016122034395027706585561529823, −7.61167792401907332400563563354, −6.98947093798112551451964301665, −5.97842134447328470281514461205, −5.25185675640393075103226568023, −4.48868532752742549152868136886, −3.32019086860507282157016380432, −2.14956752878819675939857420438, 0.20666972930885025097582369640, 1.69128829763243342862742359802, 3.87968467368243773083198099711, 4.12912279813737142109791207209, 5.02151509088163339153327338816, 6.29835344184545961971115027823, 6.98502030923373678838189253997, 7.59416077352095986142660841598, 8.600161407864128607523306815110, 10.18143349494975694079005456849

Graph of the $Z$-function along the critical line